The Phase Rule and Its Applications Part 2

Now, the results of experiment are quite in agreement with the requirements of the Phase Rule, and at any given temperature the system water--vapour can exist in equilibrium only under a definite pressure.

The vapour pressure of water at different temperatures has been subjected to careful measurement by Magnus,[22] Regnault,[23] Ramsay and Young,[24]

Juhlin,[25] Thiesen and Scheel,[26] and others. In the following table the values of the vapour pressure from -10 to +100 are those calculated from the measurements of Regnault, corrected by the measurements of Wiebe and Thiesen and Scheel;[27] those from 120 to 270 were determined {23} by Ramsay and Young, while the values of the critical pressure and temperature are those determined by Battelli.[28]

VAPOUR PRESSURE OF WATER.

-------------+-----------------+--------------+-------------------- Temperature. Pressure in cm. Temperature. Pressure in cm.

mercury. mercury.

-------------+-----------------+--------------+-------------------- -10 0.213 120 148.4 0 0.458[29] 130 201.9 +20 1.752 150 356.8 40 5.516 200 1162.5 60 14.932 250 2973.4 80 35.54 270 4110.1 100 76.00 364.3 14790.4 (194.6 atm.) (critical (critical pressure).

temperature) -------------+-----------------+--------------+--------------------

The pressure is, of course, independent of the relative or absolute volumes of the liquid and vapour; on increasing the volume at constant temperature, a certain amount of the liquid will pa.s.s into vapour, and the pressure will regain its former value. If, however, the pressure be permanently maintained at a value different from that corresponding to the temperature employed, then either all the liquid will pa.s.s into vapour, or all the vapour will pa.s.s into liquid, and we shall have either vapour alone or liquid alone.

Upper Limit of Vaporization Curve.--On continuing to add heat to water contained in a closed vessel, the pressure of the vapour will gradually increase. Since with increase of pressure the density of the vapour must increase, and since with rise of temperature the density of the liquid must decrease, a point will be reached at which the density of liquid and vapour become identical; the system ceases to be heterogeneous, and pa.s.ses into one h.o.m.ogeneous phase. The temperature at which this occurs is called the _critical temperature_. To this temperature there will, of course, correspond a certain definite pressure, called the _critical pressure_. The curve representing the {24} equilibrium between liquid and vapour must, therefore, end abruptly at the critical point. At temperatures above this point no pressure, however great, can cause the formation of the liquid phase; at temperatures above the critical point the vapour becomes a gas.

In the case of water, the critical temperature is 364.3, and the critical pressure 194.6 atm.; at the point representing these conditions the vapour-pressure curve of water must cease.

Sublimation Curve of Ice.--Vapour is given off not only by liquid water, but also by solid water, or ice. That this is so is familiar to every one through the fact that ice or snow, even at temperatures below the melting point, gradually disappears in the form of vapour. Even at temperatures considerably lower than 0, the vapour pressure of ice, although small, is quite appreciable; and it is possible, therefore, to have ice and vapour coexisting in equilibrium. When we inquire into the conditions under which such a system can exist, we see again that we are dealing with a univariant system--one component existing in two phases--and that, therefore, just as in the case of the system water and vapour, there will be for each temperature a certain definite pressure of the vapour, and this pressure will be independent of the relative or absolute amounts of the solid or vapour present, and will depend solely on the temperature. Further, just as in the case of the vapour pressure of water, the condition of equilibrium between ice and water vapour will be represented by a line or curve showing the change of pressure with the temperature. Such a curve, representing the conditions of equilibrium between a solid and its vapour, is called a _sublimation curve_. At temperatures represented by any point on this curve, the solid (ice) will sublime or pa.s.s into vapour without previously fusing. Since ice melts at 0 (_vide infra_), the sublimation curve must end at that temperature.

The following are the values of the vapour pressure of ice between 0 and -50.[30]

{25}

VAPOUR PRESSURE OF ICE.

--------------------------------------------------------------- Temperature. Pressure in mm. Temperature. Pressure in mm.

mercury. mercury.

-------------+-----------------+--------------+---------------- -50 0.050 -8 2.379 -40 0.121 -6 2.821 -30 0.312 -4 3.334 -20 0.806 -2 3.925 -15 1.279 0 4.602 -10 1.999 ----------------------------------------------------------------

Equilibrium between Ice and Water. Curve of Fusion.--There is still another univariant system of the one component water, the existence of which, at definite values of temperature and pressure, the Phase Rule allows us to predict. This is the system solid--liquid. Ice on being heated to a certain temperature melts and pa.s.ses into the liquid state; and since this system solid--liquid is univariant, there will be for each temperature a certain definite pressure at which ice and water can coexist or be in equilibrium, independently of the amounts of the two phases present. Since now the temperature at which the solid phase is in equilibrium with the liquid phase is known as the melting point or point of fusion of the solid, the curve representing the temperatures and pressures at which the solid and liquid are in equilibrium will represent the change of the melting point with the pressure. Such a curve is called the _curve of fusion_, or the melting-point curve.

It was not until the middle of the nineteenth century that this connection between the pressure and the melting point, or the change of the melting point with the pressure, was observed. The first to recognize the existence of such a relations.h.i.+p was James Thomson,[31] who in 1849 showed that from theoretical considerations such a relations.h.i.+p must exist, and predicted that in the case of ice the melting point would be lowered by pressure.

This prediction was fully confirmed by his brother, W. Thomson[32] (Lord Kelvin), who found that under a pressure {26} of 8.1 atm. the melting point of ice was -0.059; under a pressure of 16.8 atm. the melting point was -0.129.

The experiments which were first made in this connection were more of a qualitative nature, but in recent years careful measurements of the influence of pressure on the melting point of ice have been made more especially by Tammann,[33] and the results obtained by him are given in the following table and represented graphically in Fig. 2.

FUSION PRESSURE OF ICE.

--------------------------------------------------------------------- Pressure in kilogms. per Change of melting point for Temperature. sq. cm.[34] an increase of pressure of 1 kilogm. per sq. cm.

--------------------------------------------------------------------- -0 1 -2.5 336 0.0074 -5 615 0.0090 -7.5 890 0.0091 -10.0 1155 0.0094 -12.5 1410 0.0100 -15.0 1625 0.0116 -17.5 1835 0.0119 -20.0 2042 0.0121 -22.1 2200 0.0133 ---------------------------------------------------------------------

From the numbers in the table and from the figure we see that as the pressure is increased the melting point of ice is lowered; but we also observe that a very large change of pressure is required in order to produce a very small change in the melting point. The curve, therefore, is very steep. Increase of pressure by one atmosphere lowers the melting point by only 0.0076,[35] or an increase of pressure of 135 atm. is required to produce a lowering of the melting point of 1. We see further that the fusion curve bends slightly as the pressure is increased, which signifies that the variation of {27} the melting point with the pressure changes; at -15, when the pressure is 1625 kilogm. per sq. cm., increase of pressure by 1 kilogm. per sq. cm. lowers the melting point by 0.012. This curvature of the fusion curve we shall later (Chap. IV.) see to be an almost universal phenomenon.

[Ill.u.s.tration: FIG. 2.]

[Ill.u.s.tration: FIG. 3.]

Equilibrium between Ice, Water, and Vapour. The Triple Point.--On examining the vapour-pressure curves of ice and water (Fig. 3), we see that at a temperature of about 0 and under a pressure of about 4.6 mm. mercury, the two curves cut. At this point liquid water and solid ice are each in equilibrium with vapour at the same pressure. Since this is so, they must, of course, be in equilibrium {28} with one another, as experiment also shows. At this point, therefore, ice, water, and vapour can be in equilibrium, and as there are three phases present, the point is called a _triple point_.[36]

The triple point, however, does not lie exactly at 0 C., for this temperature is defined as the melting point of ice under atmospheric pressure. At the triple point, however, the pressure is equal to the vapour pressure of ice and water, and this pressure, as we see from the tables on pp. 21 and 23, is very nearly 4.6 mm., or almost 1 atm. less than in the previous case. Now, we have just seen that a change of pressure of 1 atm.

corresponds to a change of the melting point of 0.0076; the melting point of ice, therefore, when under the pressure of its own vapour, will be very nearly +0.0076, and the pressure of the vapour will be very slightly greater than 4.579 mm., which is the pressure at 0 (p. 21). The difference is, however, slight, and may be neglected here. At the temperature, then, of +0.0076, and under a pressure of 4.6 mm. of mercury, ice, water, and vapour will be in equilibrium; the point in our diagram representing this particular temperature and pressure is, therefore, the triple point of the system ice--water--vapour.

Since at the triple point we have three phases of one component, the system at this point is invariant--it possesses no degrees of freedom. If the temperature is changed, the system will undergo alteration in such a way that one of the phases will disappear, and a univariant system will result; if heat be added, ice will melt, and we shall have left water and vapour; if heat be abstracted, water will freeze, and we shall have left ice and vapour; if, when the temperature is altered, the pressure is kept constant, then we shall ultimately obtain only one phase (see Chap. IV.).

The triple point is not only the point of intersection of the vaporization and sublimation curves, but it is also the end-point of the fusion curve.

The fusion curve, as we have seen, is the curve of equilibrium between ice and water; and since at the triple point ice and water are each in equilibrium with {29} vapour of the same pressure, they must, of course, also be in equilibrium with one another.

[Ill.u.s.tration: FIG. 4.]

Bivariant Systems of Water.--If we examine Fig. 4, we see that the curves OA, OB, OC, which represent diagrammatically the conditions under which water and vapour, ice and vapour, and water and ice are in equilibrium, form the boundaries of three "fields," or areas, I., II., III. These areas, now, represent the conditions for the existence of the single phases, solid, liquid, and vapour respectively. At temperatures and pressures represented by any point in the field I., solid only can exist as a stable phase. Since we have here one component in only one phase, the system is bivariant, and at any given temperature, therefore, ice can exist under a series of pressures; and under any given pressure, at a series of temperatures, these pressures and temperatures being limited only by the curves OB, OC. Similarly also with the areas II. and III.

We see, further, that the different areas are the regions of stability of the phase common to the two curves by which the area is enclosed.[37] Thus, the phase common to the two systems {30} represented by BO (ice and vapour), and OA (water and vapour) is the vapour phase; and the area BOA is therefore the area of the vapour phase. Similarly, BOC is the area of the ice phase, and COA the area of the water phase.

Supercooled Water. Metastable State.--When heated under the ordinary atmospheric pressure, ice melts when the temperature reaches 0, and it has so far not been found possible to raise the temperature of ice above this point without liquefaction taking place. On the other hand, it has long been known that water can be cooled below zero without solidification occurring. This was first discovered in 1724 by Fahrenheit,[38] who found that water could be exposed to a temperature of -9.4 without solidifying; so soon, however, as a small particle of ice was brought in contact with the water, crystallization commenced. Superfused or supercooled water--_i.e._ water cooled below 0--is unstable only in respect of the solid phase; so long as the presence of the solid phase is carefully avoided, the water can be kept for any length of time without solidifying, and the system supercooled water and vapour behaves in every way like a stable system. A system, now, which in itself is stable, and which becomes instable only in contact with a particular phase, is said to be _metastable_, and the region throughout which this condition exists is called the metastable region. Supercooled water, therefore, is in a metastable condition. If the supercooling be carried below a certain temperature, solidification takes place spontaneously without the addition of the solid phase; the system then ceases to be metastable, and becomes _instable_.

Not only has water been cooled to temperatures considerably below the melting point of ice, but the vapour pressure of the supercooled water has been measured. It is of interest and importance, now, to see what relations.h.i.+p exists between the vapour pressure of ice and that of supercooled water at the same temperature. This relations.h.i.+p is clearly shown by the numbers in the following table,[39] and is represented in Fig.

3, {31} p. 27., and diagrammatically in Fig. 4, the vapour pressures of supercooled water being represented by the curve OA', which is the unbroken continuation of AO.

VAPOUR PRESSURE OF ICE AND OF SUPERCOOLED WATER.

--------------------------------------------------------------------- Pressure in mm. mercury.

------------------------------------------------------ Temperature. Water. Ice. Difference.

--------------------------------------------------------------------- 0 4.618 4.602 0.016[40]

-2 3.995 3.925 0.070 -4 3.450 3.334 0.116 -8 2.558 2.379 0.179 -10 2.197 1.999 0.198 -15 1.492 1.279 0.213 -20 1.005 0.806 0.199 ---------------------------------------------------------------------

At all temperatures below 0 (more correctly +0.0076), at which temperature water and ice have the same vapour pressure, the vapour pressure of supercooled water is _greater_ than that of ice at the same temperature.

From the relative positions of the curves OB and OA (Fig. 4) we see that at all temperatures above 0, the (metastable) sublimation curve of ice, if it could be obtained, would be higher than the vaporization curve of water.

This shows, therefore, that at 0 a "break" must occur in the curve of states, and that in the neighbourhood of this break the curve above that point must ascend less rapidly than the curve below the break. Since, however, the differences in the vapour pressures of supercooled water and of ice are very small, the change in the direction of the vapour-pressure curve on pa.s.sing from ice to water was at first not observed, and Regnault regarded the sublimation curve as pa.s.sing continuously into {32} the vaporization curve. The existence of a break was, however, shown by James Thomson[41] and by Kirchhoff[42] to be demanded by thermo-dynamical considerations, and the prediction of theory was afterwards realized experimentally by Ramsay and Young in their determinations of the vapour pressure of water and ice, as well as in the case of other substances.[43]

From what has just been said, we can readily understand why ice and water cannot exist in equilibrium below 0. For, suppose we have ice and water in the same closed s.p.a.ce, but not in contact with one another, then since the vapour pressure of the supercooled water is higher than that of ice, the vapour of the former must be supersaturated in contact with the latter; vapour must, therefore, condense on the ice; and in this way there will be a slow distillation from the water to the ice, until at last all the water will have disappeared, and only ice and vapour remain.[44]

Other Systems of the Substance Water.--We have thus far discussed only those systems which are const.i.tuted by the three phases--ice, water, and water vapour. It has, however, been recently found that at a low temperature and under a high pressure ordinary ice can pa.s.s into two other crystalline varieties, called by Tammann[45] ice II. and ice III., ordinary ice being ice I. According to the Phase Rule, now, since each of these solid forms const.i.tutes a separate phase (p. 9), it will be possible to have the following (and more) systems of water, in addition to those already studied, viz. water, ice I., ice II.; water, ice I., ice III.; water, ice II., ice III., forming invariant systems and existing in equilibrium only at a definite triple point; further, water, ice II.; water, ice III.; ice I., ice II.; ice I., ice III.; ice II., ice III., forming univariant systems, existing, therefore, at definite corresponding values of {33} temperature and pressure; and lastly, the bivariant systems, ice II. and ice III. Several of these systems have been investigated by Tammann. The triple point for water, ice I., ice III., lies at -22, and a pressure of 2200 kilogms. per sq. cm. (2130 atm.), as indicated in Fig. 2, p. 27.[46] In contrast with the behaviour of ordinary ice, the temperature of equilibrium in the case of water--ice II., and water--ice III., is _raised_ by increase of pressure.

B. _Sulphur._

Polymorphism.--Reference has just been made to the fact that ice can exist not only in the ordinary form, but in at least two other crystalline varieties. This phenomenon, the existence of a substance in two or more different crystalline forms, is called _polymorphism_. Polymorphism was first observed by Mitscherlich[47] in the case of sodium phosphate, and later in the case of sulphur. To these two cases others were soon added, at first of inorganic, and later of organic substances, so that polymorphism is now recognized as of very frequent occurrence indeed.[48] These various forms of a substance differ not only in crystalline shape, but also in melting point, specific gravity, and other physical properties. In the liquid state, however, the differences do not exist.

According to our definition of phases (p. 9), each of these polymorphic forms const.i.tutes a separate phase of the particular substance. As is readily apparent, the number of possible systems formed of one component may be considerably increased when that component is capable of existing in different crystalline forms. We have, therefore, to inquire what are the conditions under which different polymorphic forms can coexist, either alone or in presence of the liquid and vapour phase. For the purpose of ill.u.s.trating the general behaviour of such systems, we shall study the systems formed by the different crystalline forms of sulphur, tin, and benzophenone.

{34}

Sulphur exists in two well-known crystalline forms--rhombic, or octahedral, and monoclinic, or prismatic sulphur. Of these, the former melts at 114.5; the latter at 120.[49] Further, at the ordinary temperature, rhombic sulphur can exist unchanged, whereas, on being heated to temperatures somewhat below the melting point, it pa.s.ses into the prismatic variety. On the other hand, at temperatures above 96, prismatic sulphur can remain unchanged, whereas at the ordinary temperature it pa.s.ses slowly into the rhombic form.

If, now, we examine the case of sulphur with the help of the Phase Rule, we see that the following systems are theoretically possible:--

I. _Bivariant Systems: One component in one phase._ (_a_) Rhombic sulphur.

(_b_) Monoclinic sulphur.

(_c_) Sulphur vapour.

(_d_) Liquid sulphur.